Experimental data from the flavour sector – for instance,
neutral meson mass differences for Kaons and B-mesons, or \(\epsilon _K\)  data –
strongly constrain such FCNCs, typically forcing the extra scalars to have masses
above 1 TeV [1]}. An alternative is to fine-tune the FCNC interactions
so that they are very small, though for some models cancellations between CP-even and CP-odd
contributions to the amplitudes off the observables mentioned allow for below-TeV
scalars with a minimal fine-tuning (see, for
instance, [2]}, [3]}, [4]}). Another possibility
is to assume alignment between different Yukawa
matrices [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, though that is an ansatz
which is not preserved under renormalization [11]}. There is
yet another possibility, however: the BGL (Branco-Grimus-Lavoura)
model [12]}, [13]} is a remarkable
version of the 2HDM wherein FCNC interactions are naturally small – this results
from the imposition of a flavour-violating symmetry, i.e. a symmetry which treats
some generations of fermions differently from others. As a consequence of this symmetry,
FCNC couplings are suppressed by off-diagonal Cabibbo-Kobayashi-Maskawa (CKM) matrix
elements. The phenomenology of the model has been studied quite thoroughly (see,
for instance, [14]}, [15]}, [16]}) and it remains a valid and
exciting possibility for BSM physics.
